Number SystemsNumber Systems• Stone Age: knots, some stone marks• Roman Empire: more systematic notation I,

II, III, IV, V, VI, VII.VIII, IX, X, C=100, D=500, M=1000, L=50

• Concept of zero by – Maya- I century, Hindu-V century

• Positional-value systems: decimal, binary, octal, etc..

Positional-Value System

• The value of a digit (“digit” from Latin word for finger) depends on its position

5 6 7 . 9 1 4

MSD Decimal LSD point

Positional values 2 1 0 -1 -2 -3

(weights) 10 10 10 10 10 10

We will write ( 5 6 7. 9 1 4)10

Binary:Base-2 Number System

1 0 1 1 1 1 . 0 0 1

2 2 2 2 2 2 2 2 2 5 4 3 2 1 0 -1 -2 -3

We write: ( 1 0 1 1 1 1 . 0 0 1 )2

base point or radix

Digits are called bits

Binary Representation• The basis of all digital data is binary representation.• Binary - means ‘two’

– 1, 0– True, False– Hot, Cold– On, Off

• We must be able to handle more than just values for real world problems– 1, 0, 56– True, False, Maybe– Hot, Cold, LukeWarm, Cool– On, Off, Leaky

Number Systems

• To talk about binary data, we must first talk about number systems

• The decimal number system (base 10) you should be familiar with!– A digit in base 10 ranges from 0 to 9.– A digit in base 2 ranges from 0 to 1 (binary number system).

A digit in base 2 is also called a ‘bit’.– A digit in base R can range from 0 to R-1– A digit in Base 16 can range from 0 to 16-1

(0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). Use letters A-F to represent values 10 to 15. Base 16 is also called Hexadecimal or just ‘Hex’.

Positional Number SystemsPositional Number Systems

• The traditional number system is called a positional number system.

• A number is represented as a string of digits.

• Each digit position has a weight assoc. with it.

• Number’s value = a weighted sum of the digits

1

0

10p

ii

i

D d

410*5100*31000*66354

Positional Notation – more examples

Value of number is determined by multiplying each digit by a weight and then summing. The weight of each digit is a POWER of the BASE and is determined by position.

953.78 = 9 x 102 + 5 x 101 + 3 x 100 + 7 x 10-1 + 8 x 10-2

= 900 + 50 + 3 + .7 + .08 = 953.78

% 1011.11 = 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2

= 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.75

$ A2F = 10x162 + 2x161 + 15x160 = 10 x 256 + 2 x 16 + 15 x 1 = 2560 + 32 + 15 = 2607

decimal

binary

hex

Base 10, Base 2, Base 16The textbook uses subscripts to represent different bases (ie. A2F16 , 953.7810, 1011.112 )

We will use special symbols to represent the different bases.The default base will be decimal, no special symbol for base 10.

The ‘$’ will be used for base 16 ( $A2F)

The ‘%’ will be used for base 2 (%10101111)

If ALL numbers on a page are the same base (ie, all in base 16 or base 2 or whatever) then no symbols will be used and a statement will be present that will state the base (ie, all numbers on this page are in base 16).

Common Powers2-3 = 0.1252-2 = 0.252-1 = 0.520 = 121 = 222 = 423 = 824 = 1625 =3226 = 6427 = 12828 = 25629 = 512210 = 1024211 = 2048212 = 4096

160 = 1 = 20

161 = 16 = 24

162 = 256 = 28

163 = 4096 = 212

210 = 1024 = 1 K220 = 1048576 = 1 M (1 Megabits) = 1024 K = 210 x 210

230 = 1073741824 = 1 G (1 Gigabits)

Octal and Octal and Hexadecimal Hexadecimal

(“Hex”) Numbers(“Hex”) Numbers• Octal = base 8• Hexadecimal = base 16

– Use A – F to represent the values 10 through 16 in each position.

Usefulness of Octal and Hex Numbers• Useful for representing multi-bit binary numbers because their

radices are integer multiples of 2.

10 0101 1010 1111 . 1011 1112 = 2 5 A F . B E16

Decimal Binary Octal Hex

5 101 5 5

6 110 6 6

7 111 7 7

8 1000 10 8

9 1001 11 9

10 1010 12 A

11 1011 13 B

12 1100 14 C

13 1101 15 D

14 1110 16 E

15 1111 17 F

Comparison of binary, decimal, octal Comparison of binary, decimal, octal and hexadecimal numbersand hexadecimal numbers

examples examples of octal of octal and hex and hex numbersnumbers

Decimal to Hex Conversions

Convert 53 to HexConvert 53 to Hex

53/16 = 3, rem = 5 3 /16 = 0 , rem = 3 53 = $ 35 = 3 x 161 + 5 x 160

= 48 + 5 = 53

Hex (base 16) to Binary Conversion

Each Hex digit represents 4 bits. To convert a Hex number to Binary, simply convert each Hex digit to its four bit value.

Hex Digits to binary:$ 0 = % 0000$ 1 = % 0001$2 = % 0010$3 = % 0011$4 = % 0100$5 = % 0101$6 = % 0110$7 = % 0111$8 = % 1000

Hex Digits to binary (cont):$ 9 = % 1001$ A = % 1010$ B = % 1011$ C = % 1100$ D = % 1101$ E = % 1110$ F = % 1111

Conversions: Hex to Binary, Binary Conversions: Hex to Binary, Binary to Hexto Hex

$ A2F = % 1010 0010 1111

$ 345 = % 0011 0100 0101

Binary to Hex is just the opposite, create groups of 4 bits starting with least significant bits. If last group does not have 4 bits, then pad with zeros for unsigned numbers.

% 1010001 = % 0101 0001 = $ 51

Padded with a zero

Binary to hex Binary to hex conversionconversion

Hex to Binary Hex to Binary conversionconversion

A Trick!If faced with a large binary number that has to be converted to decimal, we first convert the binary number to HEX, then convert the HEX to decimal. Less work!

% 110111110011 = % 1101 1111 0011 = $ D F 3 = 13 x 162 + 15 x 161 + 3x160

= 13 x 256 + 15 x 16 + 3 x 1 = 3328 + 240 + 3 = 3571

Of course, you can also use the binary, hex conversion feature on your calculator. You can use calculators on exam

Bah! I thought we were talking about Binary DATA!!!

Yah, we were!

How many binary DIGITS does it take to represent our data??

Binary CodesBinary CodesOne Binary Digit (one bit) can take on values 0, 1. We can represent TWO values:

(0 = hot, 1 = cold), (1 = True, 0 = False), (1 = on, 0 = off).

Two Binary digits (two bits) can take on values of 00, 01, 10, 11. We can represent FOUR values:

(00 = hot, 01 = warm, 10 = cool, 11 = cold).

Three Binary digits (three bits) can take on values of 000, 001, 010, 011, 100, 101, 110, 111. We can represent 8 values000 = Black, 001 = Red, 010 = Pink, 011 = Yellow, 100 = Brown, 101 = Blue, 110 = Green , 111 = White.

Binary Codes (cont.)N bits (or N binary Digits) can represent 2N different values.

(for example, 4 bits can represent 24 or 16 different values)

N bits can take on unsigned decimal values from 0 to 2N-1.

Codes usually given in tabular form.

000001010011100101110111

blackredpinkyellowbrownbluegreenwhite

Code Code ConversionsConversions

( )2 ( )4 ( )8 ( )16

• To convert a binary number to a system which is base-2z, group digits together by z and convert each group separately

• 100111.1010 ---> ( )16

2 7 . A 2 7 . A

Converting from Converting from binary base hex as binary base hex as an example of base an example of base 22ZZ

Conversion of Any Base to Decimal

Converting from ANY base to decimal is done by multiplying each digit by its weight and summing.

% 1011.11 = 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2

= 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.75

Binary to Decimal

Hex to Decimal

$ A2F = 10x162 + 2x161 + 15x160 = 10 x 256 + 2 x 16 + 15 x 1 = 2560 + 32 + 15 = 2607

Conversion ( ) I ( )10

• express number as a power series in I, and add all terms using decimal addition

Converting from Converting from base I to decimalbase I to decimal

Decimal-to-Radix-r Decimal-to-Radix-r ConversionsConversions

• Radix-r-to-decimal conversions are easy since we do arithmetic in decimal.

• However, decimal-to-radix-r conversions using decimal arithmetic is harder.

• To do the latter conversion, we convert the integer and fractional parts separately and add the results afterwards.

Convert ( ) 10 ( ) r

• Integer part:– Divide the number and all successive

quotients by r– accumulate the remainders

• Fractional part:– Multiply the number and successive fractions

by r – accumulate the integers

Conversion of Decimal Integer To ANY Base

Divide Number N by base R until quotient is 0. Remainder at EACH step is a digit in base R, from Least Significant digit to Most significant digit.

Convert 53 to binary 53/2 = 26, rem = 1 26/2 = 13, rem = 0 13/2 = 6 , rem = 1 6 /2 = 3, rem = 0 3/2 = 1, rem = 1 1/2 = 0, rem = 1

53 = % 110101 = 1x25 + 1x24 + 0x23 + 1x22 + 0x21 + 1x20

= 32 + 16 + 0 + 4 + 0 + 1 = 53

Least Significant Digit

Most Significant Digit

Decimal-to-Radix-r Conversions: Integer Part

• Successively divide number by r, taking remainder as result.

• Example: Convert 5710 to binary.

57 / 2 = 28 remainder 1 (LSB)

/2 = 14 remainder 0

/2 = 7 remainder 0

/2 = 3 remainder 1

/2 = 1 remainder 1

/2 = 0 remainder 1 (MSB)

Answer: 111001Answer: 11100122

Decimal-to-Radix-r Conversions: Fractional Part

• Successively multiply number by r, taking integer part as result and chopping off integer part before next iteration.

• May be unending!

• Example: convert .310 to binary.

.3 * 2 = .6 integer part = 0

.6 * 2 = 1.2 integer part = 1

.2 * 2 = .4 integer part = 0

.4 * 2 = .8 integer part = 0

.8 * 2 = 1.6 integer part = 1

.6 * 2 = 1.2 integer part = 1, etc.

Answer = .01001

More More Conversion Conversion methods for methods for common common radicesradices

Least Significant DigitLeast Significant DigitMost Significant DigitMost Significant Digit

53 = % 110101

Most Significant Digit (has weight of 25 or 32). For base 2, also called Most Significant Bit (MSB). Always LEFTMOST digit.

Least Significant Digit (has weight of 20 or 1). For base 2, also called Least Significant Bit (LSB). Always RIGHTMOST digit.

Binary Data in your lifeBinary Data in your life

The computer screen on your Win 98 PC can be configured for different resolutions. One resolution is 600 x 800 x 8, which means that you have 600 dots vertically x 800 dots horizontally, with each dot using 8 bits to take on 256 different colors. (actually, a dot is called a pixel).

Need 8 bits to represent 256 colors ( 28 = 256). Total number of bits needed to represent the screen is then:

600 x 800 x 8 = 3,840,000 bits (or just under 4 Mbits)

Your video card must have at least this much memory on it.

1 Mbits = 1024 x 1024 = 210 x 210 = 220 . 1 Kbits = 1024 = 210.

Addition and Addition and SubtractionSubtraction

• Use same technique as decimal• Except that the addition and subtraction

tables are different• Already seen addition table

– Truth table for Sum and Cout function

Examples of decimal and Examples of decimal and corresponding binary additionscorresponding binary additions

Examples of decimal and Examples of decimal and corresponding binary subtractionscorresponding binary subtractions

Binary Addition and Subtraction TableBinary Addition and Subtraction Table

Subtraction table

bin x y bout d

0 0 0 0 0

0 0 1 1 1

0 1 0 0 1

0 1 1 0 0

1 0 0 1 1

1 0 1 1 0

1 1 0 0 0

1 1 1 1 1

borrow inborrow in borrow outborrow out

Discuss this method in comparison with previous method from the class to create a subtractor

Addition and Subtraction of Octal Addition and Subtraction of Octal and Hexadecimal Numbersand Hexadecimal Numbers

• Not really too different

• But the addition and subtraction tables must be developed.

The concept of 10’s The concept of 10’s complementcomplement

digit digit complements complements

in binary, in binary, octal, decimal octal, decimal

and and hexadecimalhexadecimal

Representation of Negative Representation of Negative NumbersNumbers

• More accurately: representation of signed numbers– Signed-magnitude representation– Radix-complement representation

• 2’s-complement representation

– Diminished radix-complement representation– Ones’ complement representation– Excess representations

Comparison of decimal and 4-bit numbers. Comparison of decimal and 4-bit numbers. ComplementsComplements

Decimal numbers, Decimal numbers, their two’s their two’s complements, complements, ones’ ones’ complements, complements, signed magniture signed magniture and excess 2and excess 2m-1m-1 binary codesbinary codes

Existence of two zeros!

EXPLAIN

Signed-magnitude Signed-magnitude representationrepresentation

• Also called, “sign-and-magnitude representation”• A number consists of a magnitude and a symbol

representing the sign• Usually 0 means positive, 1 negative

– Sign bit

– Usually the entire number is represented with 1 sign bit to the left, followed by a number of magnitude bits

Machine arithmetic with signed-magnitude representation

• Takes several steps to add a pair of numbers– Examine signs of the addends– If same, add magnitudes and give the result the same sign as

the operands– If different, must…

• Compare magnitude of the two operands• Subtract smaller number from larger• Give the result the sign of the larger operand

• For this reason the signed-magnitude representation is not as popular as one might think because of its “naturalness”

Complement number systems

• Negates a number by taking its complement instead of negating the sign

• Exact meaning of taking its complement is defined in various ways – will see

• Not natural for humans, but better for machine arithmetic

• Will describe 2 complement number systems– Radix complement – very popular in real computers– Diminished radix-complement – not very useful, may

skip or not spend much time on it

Radix-complement number representation

• Must first decide how many bits to represent the number – say n.

• Complement of a number = rn – number• Example: 4-bit decimal:

– Original number = 3524– 10’s complement = 10000-3524 = 6476

• 0 and positive numbers: 0000-4999• Negative numbers: 5000-9999, where 9999 is

‘minus 1.’

Two’s-complement Two’s-complement representationrepresentation

• Just radix-complement when radix = 2• Used a lot in computers and other digital

arithmetic circuits• 0 and positive numbers: leftmost bit = 0• Negative numbers: leftmost bit = 1• To find a number’s complement – just flip

all the bits and add 1• See graphical view – Fig. 2.3, p. 40.

Two’s-Comp Addition and Subtraction Rules

• Starting from 1000 (-8) on up, each successive 2’s comp number all the way to 0111 (+7) can be obtained by adding 1 to the previous one, ignoring any carries beyond the 4th bit position

• Since addition is just an extension of ordinary counting, 2’s comp numbers can be added by ordinary binary addition!

• No different cases based on operands’ signs!• Overflow possible

– Occurs if result is out of range– To detect – happens if operands are the same sign but sum is a

different sign of that of the operands

Modular Counting representation Modular Counting representation of Two’s Complementsof Two’s Complements

Modular Counting representation of unsigned numbersModular Counting representation of unsigned numbers

Unsigned NumbersUnsigned Numbers

Rules for addition and subtractionRules for addition and subtraction

DECIMAL DECIMAL CODESCODES

Codes for Decimal DigitsThere are even codes for representing decimal digits. These codes use 4-bits for EACH decimal digits; it is NOT the same as converting from decimal to binary.

BCD Code0 = % 00001 = % 00012 = % 00103 = % 00114 = % 01005 = % 01016 = % 01107 = % 01118 = % 10009 = % 1001

In BCD code, each decimal digit simply represented by its binary equivalent. 96 = % 1001 0110 = $ 96 (BCD code)

Advantage: easy to convertDisadvantage: takes more bits to store a number:

255 = % 1111 1111 = $ FF (binary code) 255 = % 0010 0101 0101 = $ 255 (BCD code)

takes only 8 bits in binary, takes 12 bits in BCD.

Binary code for decimal numbersBinary code for decimal numbers

• Any encoding needs at least 4 bits/decimal digit• BCD (8421), a weighted code• Packed BCD• 2421 code

– Self-complementing: the code for the 9s’ comp of any digit may be obtained by complementing the individual bits of the digit’s code word

• Excess 3– Not a weighted code, but is also self-complementing– Since code follows standard binary counting sequence,

standard binary counters can easily be made to count in excess-3

Biquinary codeBiquinary code• Uses more than 4 bits• First 2 bits indicate whether the number is in the

range 0-4 or 5-0– One-hot

• Last 5 bits indicate which of the five numbers in the selected range is represented– Also one-hot

• Advantage: error-detection property. If any 1 bit in a code word is accidentally changed to the opposite value, the resulting code word doesn’t represent a decimal digit at all – flagged as error.

Codes for CharactersCodes for CharactersAlso need to represent Characters as digital data. The ASCII code (American Standard Code for Information Interchange) is a 7-bit code for Character data. Typically 8 bits are actually used with the 8th bit being zero or used for error detection (parity checking).8 bits = 1 Byte.

‘A’ = % 01000001 = $41 ‘&’ = % 00100110 = $26

7 bits can only represent 27 different values (128). This enough to represent the Latin alphabet (A-Z, a-z, 0-9, punctuation marks, some symbols like $), but what about other symbols or other languages?

UNICODEUNICODEUNICODE is a 16-bit code for representing alphanumeric data. With 16 bits, can represent 216 or 65536 different symbols.16 bits = 2 Bytes per character.

$0041-005A A-Z$0061-4007A a-z

Some other alphabet/symbol ranges

$3400-3d2d Korean Hangul Symbols$3040-318F Hiranga, Katakana, Bopomofo, Hangul$4E00-9FFF Han (Chinese, Japenese, Korean)

UNICODE used by Web browsers, Java, most software these days.

Decimal codesDecimal codes

Always two 1’s

0+3=3.

1+3=4 etc

GRAY CODES GRAY CODES and mechanical and mechanical

encodingsencodings

A mechanical A mechanical Encoding DiskEncoding Disk

Two bits change in natural binary code

Gray Code for decimal DigitsGray Code for decimal Digits

Gray Code0 = % 00001 = % 00012 = % 00113 = % 00104 = % 01105 = % 11106 = % 10107 = % 10118 = % 10019 = % 1000

A Gray code changes by only 1 bit for adjacent values. This is also called a ‘thumbwheel’ code because a thumbwheel for choosing a decimal digit can only change to an adjacent value (4 to 5 to 6, etc) with each click of the thumbwheel. This allows the binary output of the thumbwheel to only change one bit at a time; this can help reduce circuit complexity and also reduce signal noise.

Binary vs Gray CodesBinary vs Gray Codes

You should be able to design binary to Gray code converter in both directions

A mechanical A mechanical disk for Gray disk for Gray

CodeCode

Always one bit changes only

7 bit 7 bit ASCII ASCII codecode

You should be able to design a converter in both directions from any code to any other code!

Example of sequence generator machine Example of sequence generator machine with controlling counter machinewith controlling counter machine

Controlling many devices with binary and Controlling many devices with binary and one-hot codesone-hot codes

Error Error Correcting Correcting

CodesCodes

The concept of a hypercubeThe concept of a hypercube

Even-parity and odd-parity codesEven-parity and odd-parity codes

Space of codes for 7 bits with code words and non-code wordsSpace of codes for 7 bits with code words and non-code words

8-bit, 8-bit, distance distance 4 codes4 codes

Examples of Examples of Hamming Hamming

CodesCodes

Hamming Hamming CodesCodes

7 bit 7 bit Hamming Hamming

codescodes

Extended Hamming CodesExtended Hamming Codes

Two Two dimensional dimensional

codescodes

Error-correcting code for a RAID systemError-correcting code for a RAID system

Serial data transmissionSerial data transmission

Well-known codes for serial data Well-known codes for serial data transmissiontransmission

Multiplication Multiplication and Division and Division

IntroIntro

Binary multiplicationBinary multiplication• Grammar school method for decimal: add a list of

shifted multiplicands computed according to the digits of the multiplier

• Same method can be used in binary• For two unsigned operands, the only possible

values of the multiplier digits are 0 and 1– Thus it’s trivial to form the shifted multiplicands

Binary multiplication in binary on a machine

• More convenient to add each shifted multiplicand as it is created to a partial product

• Will do an example.• In general when we multiply an n-bit number by an

m-bit number, the result requires at most n+m bits to express

• The shift-and-add algorithm requires m partial products and additions to obtain result, but the 1st addition is trivial (adding to 0)

Long DivisionLong Division

Homework problem 1Homework problem 1

Convert the following binary numbers to decimal:

•1011011.0110

•00110.11001

Homework problem 2

Convert from decimal to binary:

•0.5

•73.426

•290.9

Convert from Binary to Octal:

•1 101 011 110 111

•11 011.101 1

Homework problem 3

Homework problem 4

• Calculate 191+141 (Let’s first convert these to binary as an exercise.) Verify in decimal

• 210 – 109, calculate first binary numbers. Verify.

Homework problem 5

1. Discuss how to convert hex, binary integers to Decimal

2. Discuss how to convert decimal integers to hex, binary

3. Discuss how to convert hex to binary, binary to Hex

4. Explain why N binary digits can represent 2N values, unsigned integers 0 to 2N-1.

sources

• Bob Reese

• Wakerly

• Other from internet